Advanced Physics Textbook PDF - El Saleheya El Gadida University

 El Saleheya El Gadida University Faculty of Computers and Information ADVANCED PHYSICS By Dr. Eman Youssif Contents 1. Mechanics.................................................................................... 4 Oscillation (Periodic Motion)......................

 El Saleheya El Gadida University Faculty of Computers and Information ADVANCED PHYSICS By Dr. Eman Youssif Contents 1. Mechanics.................................................................................... 4 Oscillation (Periodic Motion)............................................................... 4 1.1. Describing oscillation........................................................... 4 1.2. Amplitude, Period, Frequency, and Angular Frequency....... 8 1.3. Simple harmonic motion....................................................... 9 1.4. Circular Motion and the Equations of SHM....................... 12 1.5. Period and amplitude in SHM............................................. 15 1.6. Displacement, Velocity, and Acceleration in SHM............ 15 1.7. Energy in Simple Harmonic Motion................................... 19 1.8. Applications of SHM........................................................... 20 1.8.1. Vertical SHM................................................................. 20 1.8.2. Angular SHM................................................................. 21 1.8.3. Vibrations of Molecules................................................. 22 1.9. The Simple Pendulum......................................................... 23 2. Mechanical waves...................................................................... 26 2.1. Types of mechanical waves................................................. 26 2.2. Periodic waves..................................................................... 28 2.2.1. Periodic transverse waves.............................................. 29 2.2.2. Periodic Longitudinal Waves......................................... 33 2.3. Mathematical Description of a Wave.................................. 36 1  2.3.1. Wave Function............................................................... 36 2.3.2. Graphing the Wave Function......................................... 38 2.3.3. Particle Velocity and Acceleration in a Sinusoidal Wave 41 2.4. Wave Intensity..................................................................... 42 2.5. Wave Reflections, Interference, Superposition.................. 44 2.6. Standing Waves on a String................................................ 49 2.7. Sound waves........................................................................ 54 2.7.1. Intensity of Sound Waves.............................................. 58 2.7.2. What Happens When Sound Travels from One Medium to Another? 59 2.7.3. Different Types of Sound Waves (Ultrasound and Shockwave) 61 2.7.4. Different Types of Shockwaves..................................... 62 3. Introduction to quantum physics................................................ 64 3.1. Atomic physics.................................................................... 64 3.2. Molecules and solids........................................................... 64 4. Lab............................................................................................. 65 4.1. Simple pendulum................................................................. 65 4.2. Hooke’s Law........................................................................ 69 4.3. Air Column Resonance........................................................ 73 2 3  1. Mechanics Oscillation (Periodic Motion) Many kinds of motion repeat themselves over and over: the vibration of a Quartz crystal in a watch, the swinging pendulum of a grandfather Clock, the sound vibrations produced by a clarinet or an organ pipe, and The back-and-forth motion of the pistons in a car engine. This kind of motion, called periodic motion or oscillation, is the subject of this chapter. Understanding periodic motion will be essential for our later study of waves, sound, alternating electric currents, and light. In this chapter we will concentrate on two simple examples of systems that can undergo periodic motions: spring-mass systems and pendulums. 1.1. Describing oscillation Figure 1 shows one of the simplest systems that can have periodic motion. The body with mass m is attached to a Spring of negligible mass that can be either stretched or compressed. The left end of the spring is held fixed and the right end is attached to the body. The spring Force is the only horizontal force acting on the body; the vertical normal and Gravitational forces always add to zero. 4  Fig. 1. Periodic motion It’s simplest to define our coordinate system so that the origin O is at the equilibrium position, where the spring is neither stretched nor compressed. Then x is the x-component of the displacement of the body from equilibrium and is also the change in the length of the spring. The x- component of the force that the spring exerts on the body is Fx and the x- component of acceleration ax is given by ax= Fx/m Figure 2 shows the body for three different displacements of the spring. Whenever the body is displaced from its equilibrium position, the spring force tends to restore it to the equilibrium position. We call a force with this character a Restoring force. Oscillation can occur only when there is a restoring force tending to return the system to equilibrium. 5 Fig. 2 model for periodic motion 6  How oscillation occurs in this system?. If we displace the body to the Right to x = A and then let go, the net force and the acceleration are to the left (Fig. 2a). The speed increases as the body approaches the equilibrium position O. When the body is at O, the net force acting on it is zero (Fig. 2b), but because of its motion it overshoots the equilibrium position. On the other side of the equilibrium position the body is still moving to the left, but the net force and the acceleration are to the right (Fig. 2c); hence the speed decreases until the body comes to a stop. We will show later that with an ideal spring, the stopping point is at x = -A. The body then accelerates to the right, overshoots equilibrium again, and stops at the starting point x = A ready to repeat the whole process. The body is oscillating! If there is no friction or other force to remove mechanical energy from the system, This motion repeats forever; the restoring force perpetually draws the body back toward the equilibrium position, only to have the body overshoot time after time. In different situations the force may depend on the displacement x from equilibrium in different ways. But oscillation always occurs if the force is a restoring Force that tends to return the system to equilibrium. 7  1.2. Amplitude, Period, Frequency, and Angular Frequency The amplitude A of the motion is the maximum magnitude of displacement from equilibrium (that is the maximum value of x ). It is a Positive value. If the spring in Fig. 2 is an ideal one, the total overall range of the motion is 2A. The SI unit of A is the meter. A complete vibration, or cycle, is one complete round trip (say: from A to –A and back to A, or from O to A, back through O to –A and back to O. Note that motion from one side to the other (say, –A to A) is a half-cycle, not a whole cycle. The period, T, is the time for one cycle. It is always positive. The SI unit is the second, but it is sometimes expressed as “seconds per cycle.” The frequency, f, is the number of cycles in a unit of time. It is always positive. The SI unit of frequency is the hertz: 1 hertz = 1 HZ = 1 cycle/s = 1 s-1 The angular frequency, ω, is 2π times the frequency: ω = 2πf why ω is a useful quantity?. It represents the rate of change of an angular quantity (not necessarily related to a rotational motion) that is 8 always measured in radians, so its units are rad/s. f is in cycle/s, the number 2π as having units rad/cycle. Relationships between frequency and period is f= T= Eq. 1 and from the definition of ω, it can be rewritten as ω = 2πf = Eq. 2 1.3. Simple harmonic motion The simplest kind of oscillation occurs when the restoring force Fx is directly proportional to the displacement from equilibrium x. The constant of proportionality between Fx and x is the force constant k. ( in Figs. 1 and 2). On either side of the equilibrium position, Fx and x always have opposite signs. It is represented that the force acting on a stretched ideal spring as The x-component of force the spring exerts on the body is the negative of this, so the x-component of force Fx on the body is Fx = -kx (restoring force exerted by an ideal spring) (Eq. 3) 9  Fig. 3 an idealized spring exerts a restoring force This equation gives the correct magnitude and sign of the force, whether x is positive, negative, or zero (Fig. 3). The force constant k is always positive and has units of N/m (a useful alternative set of units is Kg/ s2). There is no friction, so Eq. (3) gives the net force on the body. When the restoring force is directly proportional to the displacement from equilibrium, as given by Eq. (3), the oscillation is called simple harmonic motion, abbreviated SHM. The acceleration 10  ax = d2x/dt2 = Fx/m of a body in SHM is given by (simple harmonic motion) Eq. 4 The minus sign means the acceleration and displacement always have opposite signs. A body that undergoes simple harmonic motion is called a harmonic oscillator. Why is simple harmonic motion important? Keep in mind that not all periodic motions are simple harmonic; in periodic motion in general, the restoring force depends on displacement in a more complicated way than in Eq. 3. But in many systems the restoring force is approximately proportional to displacement if the displacement is sufficiently small (Fig. 4). That is, if the amplitude is small enough, the oscillations of such systems are approximately simple harmonic and therefore approximately described by Eq. 4. Thus we can use SHM as an approximate model for many different periodic motions, such as the vibration of the quartz crystal in a watch, the motion of a tuning fork, the electric current in an alternating-current circuit, and the oscillations of atoms in molecules and solids. 11  Fig. 4 Real oscillations Hooke’s low 1.4. Circular Motion and the Equations of SHM To explore the properties of simple harmonic motion, we must express the displacement x of the oscillating body as a function of time, x(t). The second derivative of this function, must be equal to times the function itself, as required by Eq. 4. 12  Figure 5a shows a top view of a horizontal disk of radius A with a ball attached to its rim at point Q. The disk rotates with constant angular speed ω (measured in rad/s), so the ball moves in uniform circular motion. A horizontal light beam shines on the rotating disk and casts a shadow of the ball on a screen. The shadow at point P oscillates back and forth as the ball moves in a circle. We then arrange a body attached to an ideal spring, like the combination shown in Figs. 1 and 2, so that the body oscillates parallel to the shadow. We will prove that the motion of the body and the motion of the ball’s shadow are identical if the amplitude of the body’s oscillation is equal to the disk radius A, and if the angular frequency 2πf of the oscillating body is equal to the angular speed ω of the rotating disk. That is, simple harmonic motion is the projection of uniform circular motion onto a diameter. The circle in which the ball moves so that its projection matches the motion of the oscillating body is called the reference circle; we will call the point Q the reference point. We take the reference circle to lie in the xy- plane, with the origin O at the center of the circle (Fig. 5b). At time t the vector OQ from the origin to the reference point Q makes an angle θ with the positive x-axis. As the point Q moves around the reference circle with 13 constant angular speed ω the vector OQ rotates with the same angular speed. Such a rotating vector is called a phasor. Fig. 5 Circular motion For the acceleration of a harmonic oscillator, provided that the angular speed ω of the reference point Q is related to the force constant k and mass m of the oscillating body by ω2 = ω=√ SHM Using the same symbol for the angular speed ω of the reference point Q and the angular frequency of the oscillating point P. The reason is that these quantities are equal! If point Q makes one complete revolution in time T, then point P goes through one complete cycle of oscillation in the same 14 time; hence T is the period of the oscillation. During time T the point Q moves through radians 2π, so its angular speed is ω = 2 π/T. NOTE Don’t confuse frequency f and angular frequency ω , You can run into trouble if You don’t make the distinction between frequency ƒ and angular frequency ω = 2 πf. Frequency tells you how many cycles of oscillation occur per second, while angular Frequency tells you how many radians per second this corresponds to on the reference circle. 1.5. Period and amplitude in SHM Eqs. 11 and 12 show that the period and frequency of SHM are completely determined by the mass m and the force constant k. In simple harmonic motion the period and frequency do not depend on the amplitude A. For given values of m and k, the time of one complete oscillation is the same whether the amplitude is large or small. If you encounter an oscillating body with a period that does depend on the amplitude, the oscillation is not simple harmonic motion. 1.6. Displacement, Velocity, and Acceleration in SHM We still need to find the displacement x as a function of time for a harmonic oscillator. For a body in SHM along the x-axis is identical to Eq. 8 for the x-coordinate of the reference point in uniform circular motion with 15 constant angular speed ω = √. Eq. 5, x = A cosθ describes the x-coordinate for both of these situations. If at t= 0, the phasor OQ makes an angle Φ with the positive x-axis, then at any later time t this angle is θ = ωt+ Φ. In such case, Eq. 5 is x = A cos(ωt+ Φ) Displacement in SHM Eq.3 Figure 9 shows a graph of Eq.13 for the particular case Φ=0. The displacement x is a periodic function of time, as expected for SHM. The value of the cosine function is always between -1 and 1, so in Eq. 13, x is always between –A and A. This confirms that A is the amplitude of the motion. Fig. 9: SHM 16  The period T is the time for one complete cycle of oscillation, as shown in Fig. 9. The cosine function repeats itself whenever the quantity in parentheses in Eq. 13 increases by 2π radians. Thus, if we start at time t = 0 the time T to complete one cycle is given by ω T = √ T = 2π or T =2π √ We find the velocity vx and acceleration ax as functions of time for a harmonic oscillator by taking derivatives of Eq..13 with respect to time: vx = = - ωA sin(ωt+ Φ) velocity in SHM ax = = = - ω2A cos(ωt+ Φ) acceleration in SHM As in cleared in Fig. 12., the velocity vx oscillates between vmax = + ωA and - vmax = - ωA and the acceleration ax oscillates between amax = +ω2A and -amax = - ω2A. 17  Fig. 12. velocity vx oscillates between vmax = + ωA and - vmax = - ωA and the acceleration ax oscillates between amax = +ω2A and -amax = - ω2A. 18  1.7. Energy in Simple Harmonic Motion Take another look at the body oscillating on the end of a spring in Figs. 2 and 13. We’ve already noted that the spring force is the only horizontal force on the body. The force exerted by an ideal spring is a conservative force, and the vertical forces do no work, so the total mechanical energy of the system is conserved. We also assume that the mass of the spring itself is negligible. The kinetic energy of the body is K = mv2 and the potential energy of the spring is U = kx2. The total mechanical energy E is also directly related to the amplitude A of the motion. When the body reaches the point x = A or – A and vx = 0, its maximum displacement from equilibrium and at this point the energy is entirely potential, and E = U = kA2. Because E is constant, it is equal to kA2 at any other point. Eq. 21 is total energy in SHM. E = mv2 + kx2 = kA2 = constant total mechanical energy in SHM Figure 15 is a graphical display of Eq. 21; energy (kinetic, potential, and total) is plotted vertically and the coordinate x is plotted horizontally. 19  Fig. 15. kinetic, potential, and total mechanical energy 1.8. Applications of SHM 1.8.1. Vertical SHM Fig. 17a, a spring with force constant k and suspend from it a body with mass m, oscillations will now be vertical. Fig. 17b, the body hangs at rest, in equilibrium. In this position the spring is stretched an amount Δl just great enough that the spring’s upward vertical force kΔl on the body balances its weight mg: kΔl = mg Take x = 0 to be this equilibrium position and take the positive x- direction to be upward. When the body is a distance x above its equilibrium 20 position (Fig. 17c), the extension of the spring is Δl-x. The net x-component of force on the body is Fnet= k(Δl- x) + (- mg)= -kx Fig. 17 : a mass attached to a hanging spring 1.8.2. Angular SHM A mechanical watch keeps time based on the oscillations of a balance wheel (Fig. 18). The wheel has a moment of inertia I about its axis. It’s a good thing that the motion of a balance wheel is SHM. If it weren’t, the frequency might depend on the amplitude, and the watch would run too fast or too slow as the spring run down. 21  Fig. 18: the balance wheel of a mechanical watch. 1.8.3. Vibrations of Molecules When two atoms are separated from each other by a few atomic diameters, as seen in fig. 19, they can exert attractive forces on each other. But if the atoms are so close to each other that their electron shells overlap, the forces between the atoms are repulsive. Between these limits, there can be an equilibrium separation distance at which two atoms form a molecule. If these atoms are displaced slightly from equilibrium, they will oscillate. As an example, we’ll consider one type of interaction between atoms called the van der Waals interaction. 22  Fig. 19: Two-atom system 1.9. The Simple Pendulum A simple pendulum is an idealized model consisting of a point mass suspended by a massless, unstretchable string. When the point mass is pulled to one side of its straight-down equilibrium position and released, it oscillates about the equilibrium position. Fig. 20a can be modeled as simple pendulums. The path of the point mass is not a straight line but the arc of a circle with radius L equal to the length of the string (Fig. 20b). We use as our coordinate the distance x 23 measured along the arc. If the motion is simple harmonic, the restoring force must be directly proportional to x. The restoring force Fθ is given in Eq. 30 Fθ = -mgsinθ The restoring force Fθ is provided by gravity; the tension T merely acts to make the point mass move in an arc. The restoring force is proportional not to θ but to sinθ, so the motion is not simple harmonic. However, if the angle θ is small, sinθ is very nearly equal to in radians. For example, θ=0.1 rad (about 6o) sinθ= 0.0998. Fθ = -mgθ = -mg The angular frequency ω of a simple pendulum with small amplitude is ω=√ = √ =√ The corresponding frequency f is f= = √ and period is T= √ 24 Fig. 20: simple pendulum 25  2. Mechanical waves Seismic tremors triggered by an earthquake, musical sounds and ripples on a pond all these are wave phenomena. Waves can occur whenever a system is disturbed from equilibrium and when the disturbance can travel, or propagate, from one region of the system to another. As a wave propagates, it carries energy. The energy in light waves from the sun warms the surface of our planet; the energy in seismic waves can crack our planet’s crust. Mechanical waves travel within some material called a medium. Not all waves are mechanical in nature. Electromagnetic waves such as light, radio waves, infrared and ultraviolet radiation, and x rays can propagate even in empty space, where there is no medium. 2.1. Types of mechanical waves As previously referred, mechanical wave is a disturbance that travels through medium. As the wave travels through the medium, the particles that make up the medium undergo displacements of various kinds, depending on the nature of the wave. Transverse wave, In Fig. 21a, the medium is a string or rope under tension. If we give the left end a small upward shake or wiggle, the wiggle travels along the length of the string. Successive sections of string go 26 through the same motion that we gave to the end, but at successively later times. The displacements of the medium are perpendicular or transverse to the direction of travel of the wave along the medium. Fig. 21: Three ways to make a wave Longitudinal wave, In Fig. 21b, the medium is a liquid or gas in a tube with a rigid wall at the right end and a movable piston at the left end. If we give the piston a single back-and-forth motion, displacement and pressure fluctuations travel down the length of the medium. This time the motions of the particles of the medium are back and forth along the same direction that the wave travels. Combination longitudinal and transverse wave, In Fig. 21c, the medium is a liquid in a channel, such as water in a canal. When we move the 27 flat board at the left end forward and back once, a wave disturbance travels down the length of the channel. In this case the displacements of the water have both longitudinal and transverse components. It must be taken into consideration three things in common. First, in each case the disturbance travels or propagates with a definite speed through the medium. This speed is called the speed of propagation, or wave speed v. Its value is determined in each case by the mechanical properties of the medium. Second, the medium itself does not travel through space; its individual particles undergo back-and-forth or up-and-down motions around their equilibrium positions. Third, to set any of these systems into motion, we have to put in energy by doing mechanical work on the system. The wave motion transports this energy from one region of the medium to another. Waves transport energy, but not matter, from one region to another. 2.2. Periodic waves The transverse wave on a stretched string in Fig. 21a is an example of a wave pulse. The hand shakes the string up and down just once, exerting a transverse force on it as it does so. The result is a single pulse, that travels 28 along the length of the string. The tension in the string restores its straight line shape once the pulse has passed. A more interesting situation develops when we give the free end of the string a repetitive, or periodic, motion. Then each particle in the string also undergoes periodic motion as the wave propagates, and we have a periodic wave. 2.2.1. Periodic transverse waves Figure 22 shows the wave that results is a symmetrical sequence of crests and troughs. Periodic waves with simple harmonic motion are particularly easy to analyze; it is called sinusoidal waves. Fig. 22: sinusoidal waves Figure 23 shows the shape of a part of the string near the left end at time intervals of 1/8 of a period, for a total time of one period. The wave 29 shape advances steadily toward the right, as indicated by the highlighted area. As the wave moves, any point on the string (any of the red dots, for example) oscillates up and down about its equilibrium position with simple harmonic motion. When a sinusoidal wave passes through a medium, every particle in the medium undergoes simple harmonic motion with the same frequency. 30 Fig. 23: the shape of a part of the string near the left end at time intervals of 1/8 of a period, for a total time of one period. 31  It is must be distinguished between the motion of the transverse wave along the string and the motion of a particle of the string. The wave moves with constant speed along the length of the string, while the motion of the particle is simple harmonic and transverse (perpendicular) to the length of the string. For a periodic wave, the shape of the string at any instant is a repeating pattern. The length of one complete wave pattern is the distance from one crest to the next, or from one trough to the next, or from any point to the corresponding point on the next repetition of the wave shape. This distance is called wavelength and denoted by λ. The wave pattern travels with constant speed v and advances a distance of one wavelength in a time interval of one period T. All points on the string oscillate with the same frequency f. The amplitude A is the height of a crest above the equilibrium level. Fig. 23 Waves on a string propagate in just one dimension. v= λf or v= λ/T periodic wave Figure 24 shows a wave propagating in two dimensions on the surface of a tank of water. 32 Fig. 24: a wave propagating in two dimensions on the surface of a tank of water 2.2.2. Periodic Longitudinal Waves Fig. 21b illustrates the mechanics of a periodic longitudinal wave. A long tube filled with a fluid, with a piston at the left end. If we push the piston in, we compress the fluid near the piston, increasing the pressure in this region. This region then pushes against the neighboring region of fluid, and so on, and a wave pulse moves along the tube. As shown in Fig. 25, this motion forms regions in the fluid where the pressure and density are greater or less than the equilibrium values. A region of increased density (darkly shaded areas) is compression and a region of reduced density (lightly shaded areas) is a rarefaction. 33 Fig. 25: using an oscillating piston to make a sinusoidal longitudinal wave The wave propagating in the fluid-filled tube at time intervals of 1/8 of a period, for a total time of one period as in Figure 26. Each particle in the fluid oscillates in SHM parallel to the direction of wave propagation with the same amplitude A and period T as the piston. The equation v= λf holds for longitudinal waves as well as for transverse waves, and indeed for all types of periodic waves. 34 Fig. 26: The wave propagating in the fluid-filled tube at time intervals of 1/8 of a period, for a total time of one period 35  2.3. Mathematical Description of a Wave Waves on a string are transverse; during wave motion a particle with equilibrium position x is displaced some distance y in the direction perpendicular to the x-axis. The value of y depends on which particle we are talking about (that is, y depends on x) and also on the time t when we look at it. Thus y is a function of both x and t; y(x, t). y= y(x, t) the wave function that describes the wave. 2.3.1. Wave Function The particle at point B in Fig. 27 is at its maximum positive value of y at t = 0 and returns to y = 0 at t = 2/8 T. these same events occur for a particle at point A or point C at t = 4/8 T and t = 6/8 T exactly one half- period later. Suppose that the displacement of a particle at the left end of the string (x = 0), the wave originates, is given by: y(x = 0, t) = A cos ωt = A cos 2πft --- It means that the particle oscillates in simple harmonic motion with amplitude A, frequency f and angular frequency ω =2πf. The wave disturbance travels from x = 0 to some point x to the right of the origin, time given by x/v. the motion of point x at time t is the same as the motion of point x = 0 at the earlier time t- x/v. 36 Fig. 27: tracking the oscillations of 3 points on a string 37  Hence we can find the displacement of point x at time t by Eq. y(x, t) = A cos 2πf [t- (x/v)] =A cos ω[t- (x/v)] cos (-θ)= cos (+θ) this Eq. can be rewritten y(x, t) = A cos ω[(x/v)- t] sinusoidal wave moving in + x-direction Eq. can be expressed in several different forms. We can express it in terms of the period T = 1/f and wave length λ = v/f. y(x, t) = A cos 2πf [(x/v)- t] = A cos [2π(x/ λ )- (t/T)] To define a quantity k, called the wave number: K = 2π / λ wave number y(x, t) = A cos [kx - ω t] ω = kv periodic wave 2.3.2. Graphing the Wave Function The wave function as a function of x for a specific time t. Figure 28a graph gives the displacement y of a particle from its equilibrium position as a function of the coordinate x of the particle. In particular, at time t = 0 38  y(x, t=0 ) = A cos kx = A cos 2π (x / λ) Figure 28b is a graph of the wave function versus time t for a specific coordinate x. This graph gives the displacement y of the particle at that coordinate as a function of time; that is, it describes the motion of that particle. In particular, at the position x = 0 y(x=0, t ) = A cos ωt = A cos2π (t/T) 39 Fig. 28: The wave function as a function of x for a specific time t 40  2.3.3. Particle Velocity and Acceleration in a Sinusoidal Wave From the wave function we can get an expression for the transverse velocity vy of any particle in a transverse wave. This vy to distinguish it from the wave propagation speed v. Transverse velocity vy at a particular point x can be found by taking derivative of the wave function y(x, t) with respect to t, keeping x constant (because we are looking at a particular point on the string) y(x, t) = A cos [kx - ω t] wave function vy (x, t) = ω A sin [kx - ω t] transverse velocity of a particle varies with time The maximum particle speed is ω A this can be greater than, less than, or equal to the wave speed v depending on the amplitude and frequency of the wave. The acceleration of any particle (ay ) is the second partial derivative of y(x, t) with respect to t: ay (x, t) = - ω2 A cos [kx - ω t] = - ω2 y(x, t) 1 41 The acceleration of a particle equals - ω2 times its displacement. The second partial derivative with respect to x is the curvature of the string: = -k2A cos [kx - ω t] = -k2 y(x, t) 2 From Eqs. 1 and 2 and ω = kv ω2/ k2 = v2 = Wave equation Wave equation is defined as one of the most important equations in mechanics that describes the movement of strings, wires, and fluid surfaces like water waves. 2.4. Wave Intensity Waves on a string carry energy in just one dimension of space (along the direction of the string). But other types of waves, including sound waves in air and seismic waves in the body of the earth, carry energy across all three dimensions of space. For waves that travel in three dimensions, intensity (I) is the time average rate at which energy is transported by the wave, per unit area, across a surface perpendicular to the direction of 42 propagation. That is, intensity I is average power per unit area. It is usually measured in watts per square meter (W/m2). If waves spread out equally in all directions from a source, the intensity at a distance r from the source is inversely proportional to r2. If the power output of the source is P, then the average intensity I1 through a sphere with radius r1 and surface area 4π r21 I1 = P/4π r21 The average intensity I2 through a sphere with a different radius r2 is given by a similar expression. If no energy is absorbed between the two spheres, the power P must be the same for both, and Inverse-square law for intensity is I1/ I2 = r22/ r21 The intensity I at any distance r is therefore inversely proportional to r2, Fig. 29. 43 Fig. 29: wave intensity 2.5. Wave Reflections, Interference, Superposition Waves propagate continuously in the same direction, but when a wave strikes the boundaries of its medium, all or part of the wave is reflected. The sound wave is reflected from the rigid surface and you hear an echo. In case of interference, the initial and reflected waves overlap in the same region of the medium. In general, the term “interference” refers to 44 what happens when two or more waves pass through the same region at the same time. As a simple example of wave reflections and the role of the boundary of a wave medium, let’s look again at transverse waves on a stretched string. What happens when a wave pulse or a sinusoidal wave arrives at the end of the string? If the end is fixed, The arriving wave exerts a force on the support; the reaction to this force, exerted by the support on the string, “kicks back” on the string and sets up a reflected pulse or wave traveling in the reverse direction (opposite direction from the initial). In case of a free end, as in the string might be tied to a light ring that slides on a frictionless rod perpendicular to the string. When a wave arrives at this free end, the ring slides along the rod. The ring reaches a maximum displacement, and both it and the string come momentarily to rest, as in drawing 4 in Fig. 30b. But the string is now stretched, giving increased tension, so the free end of the string is pulled back down, and again a reflected pulse is produced (drawing 7). The direction of reflected pulse is the same as for the initial pulse. The conditions at the end of the string, such 45 as a rigid support or the complete absence of transverse force, are called boundary conditions. Fig. 30: wave reflection Superposition is occurred by combining the displacements of the separate pulses at each point to obtain the actual displacement. When two waves overlap, the actual displacement of any point on the string at any time is obtained by adding the displacement the point would have if only the first wave were present and the displacement it would have if only the second 46 wave were present. In other words, the wave function that describes the resulting motion in this situation is obtained by adding the two wave functions for the two separate waves: y(x, t) = y1(x, t) + y2(x, t) The principle of superposition is of central importance in all types of waves. When a friend talks to you while you are listening to music, you can distinguish the sound of speech and the sound of music from each other. This is precisely because the total sound wave reaching your ears is the algebraic sum of the wave produced by your friend’s voice and the wave produced by the speakers of your stereo. If two sound waves did not combine in this simple linear way, the sound you would hear in this situation would be a hopeless jumble. Superposition also applies to electromagnetic waves (such as light) and many other types of waves, Fig. 31. 47 Fig. 31: wave Superposition 48  2.6. Standing Waves on a String Figure 32 shows a string that is fixed at its left end. Its right end is moved up and down in simple harmonic motion to produce a wave that travels to the left; the wave reflected from the fixed end travels to the right. The resulting motion when the two waves combine no longer looks like two waves traveling in opposite directions. The string appears to be subdivided into a number of segments, as in the time-exposure photographs of Figs. 32a_d. Figure 32e shows two instantaneous shapes of the string in Fig. 32b. In a wave that travels along the string, the amplitude is constant and the wave pattern moves with a speed equal to the wave speed. Here, instead, the wave pattern remains in the same position along the string and its amplitude fluctuates. There are particular points called nodes (labeled N in Fig. 32e) that never move at all. Midway between the nodes are points called antinodes (labeled A in Fig. 32e) where the amplitude of motion is greatest. Because the wave pattern doesn’t appear to be moving in either direction along the string, it is called a standing wave. (To emphasize the difference, a wave that does move along the string is called a traveling wave.) 49 50 51 Figure 32: shows a string that is fixed at its left end. The principle of superposition explains how the incident and reflected waves combine to form a standing wave. At certain instants, such as (t = ¼ T ), the two wave patterns are exactly in phase with each other, and the shape of the string is a sine curve with twice the amplitude of either individual wave. At the antinodes the displacements of the two waves are always identical, giving a large resultant displacement; this phenomenon is called constructive interference. At other instants, such as (t = ½ T ), the two waves are exactly out of phase with each other, and the total wave at that instant is zero. The resultant displacement is always zero. These are the nodes. At a node the displacements of the two waves are always equal and opposite and cancel each other out. This cancellation is called destructive interference. 52  As shown in Fig. 32, the distance between successive nodes or between successive antinodes is one half-wavelength ( ½ λ). The wave function for the standing wave: y(x, t) = y1(x, t) + y2(x, t) = ASW sin kx (sin ωt) The standing-wave amplitude ASW is ASW = 2A To find the positions of the nodes; these are the points for which sin kx =0, so the displacement is always zero. This occurs when kx = 0, π, 2π, 3 π,…. Or using k=2π/λ x = 0, π/k, 2π/k, 3π/k,…. x = 0, λ/2, 2λ/2, 3λ/2,…. In particular, there is a node at x = 0 , as there should be, since this point is a fixed end of the string. A standing wave, unlike a traveling wave, does not transfer energy from one end to the other. The two waves that form it would individually carry equal amounts of power in opposite directions. There is a local flow of energy from each node to the adjacent antinodes and back, but the average rate of energy transfer is zero at every point. 53  2.7. Sound waves Wave is a disturbance in a medium that transports energy without permanently transporting matter. Sound wave is a mechanical wave, that means propagation of a series of compressions followed by relaxations of particles of a medium such as gas, liquid, solid. A wave is a periodic disturbance that travels in medium, say in the x direction. It is periodic in space, which means that at any given time t, the disturbance repeats periodically with x, as in Fig. 33. It is periodic in time, which means that at any given position x, the disturbance repeats periodically with time t. The disturbance travels with a speed v, the speed of sound, so from time t1 to time t2 the disturbance travels a distance δx = x2 − x1 = v(t2 − t1). The quantity x − vt does not change for the disturbance as it “travels” with the wave. Sound waves are often described by their frequency and amplitude (figure 33). Frequency is the number of times a disturbance is propagated during a specified time interval. Hertz (Hz) is the international system of unit for frequency, which is equal to the number of repeated compressions and relaxations per second. Amplitude is the maximum displacement from a neutral starting point of a repeating event. 54  Sound waves in gases and liquids are longitudinal or compressional in that these changes occur in the same direction as the wave propagates, here in the x direction. If you pluck a string, the wave propagates along the string, but the actual disturbance of the string is perpendicular to it, making it a transverse wave. (In solids, sound waves can be longitudinal or transverse.) The Speed and Properties of Sound Waves Sound wave is the speed of sound in a specific medium. This is independent of frequency or amplitude and is defined by the velocity (or distance traveled per unit of time) of a sound wave in a specific medium. The speed of sound in a medium is affected by the intrinsic stiffness (at a specific temperature) and the density of the medium through which the sound is being propagated. Sound waves move at a speed vs that is determined by the properties of the medium. In general the sound speed is vs = √ C is a constant describing the stiffness of the material. p is the mass density or molecular density. 55 56 Fig. 33: Sound waves are compressional waves Pressure P and density variations are in phase with each other, meaning that they both increase (compression) or decrease (rarefraction) from the ambient values together. The equations of state of materials, P = nRT for ideal gases, The speed of sound in gases is vs = √ =√ γ is the ratio of the specific heats at constant pressure (cp) and volume (cv). R is the constant in the ideal gas law (R = 8.31 J/mol-K), m is the molecular mass. 57 The speed of sound in air is 343 m/s (at 20 ◦C), which is 15× slower than that in steel, while in water it is 1,482 m/s. Waves are periodic in at one t and repeat with a spatial periodicity called the wavelength λ. Wavelength is the distance between repetitions of a wave. They are also periodic in time at one x with a temporal periodicity called the period T , which corresponds to a frequency f with f = 1/T. Frequency f has units of cycles per second (cps) = Hertz (Hz). Sometimes we will use the radial frequency ω, which has units of rad/s or just 1/s, and which is related to the frequency by ω = 2π f. The wavelength, frequency, and speed of a wave are interrelated by vs = λ f. 2.7.1. Intensity of Sound Waves The intensity I of a sound wave is the energy carried by the wave per unit area and per unit time (in units of J/m2-s or W/m2). At a distance R from an isotropic source of average acoustic power P power, the intensity is I = Ppower/4πR2. 58  The acoustic impedance of amedium Z is given by the product of the mass density and sound speed for that medium, so Z = ρvs. 2.7.2. What Happens When Sound Travels from One Medium to Another? In Fig. 34 shows sound wave travels in medium 1 with intensity Ii and pressure Pi incident on the interface. The part that is reflected back into the same medium has intensity Ir and pressure Pr, and the part that is transmitted into medium 2 has intensity It and pressure Pt. These pressures are related by “matching the boundary conditions” at the interface. 59 In Fig. 34: sound wave travels in medium 1 with intensity Ii and pressure Pi incident on the interface. The magnitudes of the pressures must match so there is no net force on the interface. This gives Pi + Pr = Pt. For a wave initially moving to the right it is positive for the incident and transmitted waves and negative for the reflected wave. The fraction of intensity that is reflected is 60  Rref = = The fraction of intensity that is transmitted is Ttrans = = Ii = Ir + It and Rrefl + Ttrans = 1, which means that sound energy (and intensity) is conserved. 2.7.3. Different Types of Sound Waves (Ultrasound and Shockwave) Sound waves have many important applications in medicine. Audible sound waves must have a sufficient amplitude, and appropriate frequency (between approximately 20–20,000 Hz) to be detected by the human ear. For similar waves with frequencies above range of human hearing is called ultrasonic and below is infrasonic. An ultrasound wave is a wave that has a frequency above 20,000Hz, and is therefore not detected by the human ear. However, these high frequency waves are useful because different tissues reflect ultrasound waves differently. In ultrasonography, the reflected ultrasound waves are recorded by transducers and translated into a recorded image. 61  A shockwave is generated when a wave propagates through a medium at a speed faster than the speed of sound travels through that medium. Shockwaves produce an abrupt spike in pressure over a very short time period, as seen in Fig. 35. The total energy of the wave is derived from the area under the curve of pressure over time. In shockwave therapy, energy flux density to refers to the amount of energy delivered to a specific area of tissue, and therefore can be modulated by changing the peak pressure or area targeted. Fig. 35: shockwave 2.7.4. Different Types of Shockwaves Shockwaves are classically generated by three different types of energy sources: electrohydraulic, electromagnetic, or piezoelectric. 62  Electrohydraulic shockwave generators, tips of an electrode are submerged in a fluid. When voltage is applied, the fluid is vaporized, which causes rapid expansion in the surrounding fluid leading to shockwave propagation. Electromagnetic shockwave generators , a fluid is disturbed by applying a voltage across metallic membranes to produce a magnetic field that causes an abrupt movement in a metallic membrane and corresponding shockwave propagation. Pizoelectric shockwave generators, Fig. 36, piezoceramic elements are embedded into a spherical device submerged in a medium; when voltage is applied the ceramic elements expand, leading to a mechanical disturbance in the surrounding medium. Fig. 36: Pizoelectric shockwave generators 63 3. Introduction to quantum physics 3.1. Atomic physics 3.2. Molecules and solids 64  4. Lab. 4.1. Simple pendulum Aim Using a Simple Pendulum to determine the gravitational acceleration. Theory The simple pendulum is another mechanical system that moves in an oscillatory motion. It consists of a point mass m suspended by means of light inextensible string of length L from a fixed support as shown in Fig. 1. The motion occurs in a vertical plane and is driven by a gravitational force. The forces which are acting on the mass are shown in the figure. The tangential component of the gravitational force, mgsinθ, always acts towards the mean position θ = 0 opposite to the displacement, restoring force acting tangent to the arc. 65  The angular frequency ω of a simple pendulum with small amplitude is ω=√ = √ =√. The corresponding frequency f is f= √ and time period is T √. where L is the length of the pendulum, and g is the acceleration due to gravity at the place of experiment. Gravitational acceleration is g = 4π2(L/T2). Apparatus and material required Clamp stand; a split cork; a heavy metallic (brass/iron) spherical bob with a hook; a long, fine, strong cotton thread/string (about 2.0 m); stop- watch; metre scale, graph paper, pencil, eraser. Method 66  Place the clamp stand on the table. Tie the hook, attached to the pendulum bob, to one end of the string of about 150 cm in length. Pass the other end of the string through two half-pieces of a split cork. 1. Determine length of thread to center bob 2. Chang length of thread and displace the pendulum and record time period of 20 period using stop-watch. 3. Repeat these steps for different length pendulum. 4. Record the data in table 5. Relationship between L and T2 Result L (cm) Time of 20 Time of T2 period s one period T (s) 45 67 g = 4π2(L/T2). Or g = 4π2 slope. 68  4.2. Hooke’s Law Aim In this experiment, you will observe that the restoring force of the spring is indeed directly proportional to the amount of its stretch or compression, determine the spring constant, investigate how the period of oscillations depends on mass attached to the spring, observe oscillation patterns, and compare your experimental results with theoretical predictions. Theory Hooke’s law relates the force that a spring exerts on a body to the stretch of the spring. A relationship between stress (the force applied to a spring) and strain (the stretching that results from the stress applied). Fx = - kx Fx is the force exerted by the spring x=xf-xi is the absolute value of the difference between a length of a stretched or compressed spring. 69  k is the coefficient of proportionality and it is called the spring constant, which is unique for each spring and describes how easy or hard to stretch or to compress the spring. Minus sign in the equation reflects the fact that the restoring force tries to bring the spring back to its equilibrium position: compressed spring “pushes back” and stretched spring “pulls back”. If the restoring force of the spring is the net force acting on a mass attached, then F =- kx= mg Apparatus Spring, mass hanger, masses, meter stick, support rods. 70  Procedure 1. Determine the initial length x1 2. Add a 50 gram mass to the mass hanger (trial 1), and then increase the applied mass by 50 grams for each trial. 3. Determine the final length x2 4. Record the data in table 5. Relationship between Δx and mass m Result Slop = Δm/ Δx K = (Δm/ Δx)g = (slop)g Mass m x1 x2 x =( x1+ (gm) x2)/2 (m) 71 Relationship between Δx and mass m 72  4.3. Air Column Resonance Aim Measuring sound speed by air column resonance Introduction In this experiment, you will be using different frequencies produced by a frequency generator interface to determine the velocity of sound in air. Since the frequencies used will be known, to calculate the velocity of sound you need to find the wavelength corresponding to that sound. By holding the speaker connected to the frequency generator over a variable length tube, the sound waves will travel down the air column in the tube, and once reaching the bottom, they will be reflected back up. If the length of the air column is such that the returning waves are in phase with those being produced by the generator, they will reinforce each other and produce a noticeable increase in the sound level known as resonance. By manipulating the water level in the tube, you can change the length of the air column (L). Resonance occurs if this length of the air column is an odd number of a quarter wavelength (λ) according to the frequency being used (when L=1/4 λ, 3/4 λ, 5/4 λ, etc). 73  Apparatus  Tuning fork  Resonance tube with water reservoir 74  Theory The changes in the displacement of the air can be modeled as a sine function, with the node producing the least displacement and the antinode producing the greatest. At the closed end of the air column (at the water level), the air molecules cannot move freely, but at the open end they are free to do so. Resonance will take place when the sound’s wavelength fits into the air column such that there is a node at the closed end and an antinode at the open end. There is a small adjustment to make to the length, because the antinode is not exactly at the open end of the tube but rather slightly above it. This distance is mainly dependent on the diameter of the tube and is called the End Correction Factor (𝐸𝐶𝐹). It is equal to approximately 0.6 of the radius of the tube. 𝐸𝐶𝐹 = 𝑅𝑎𝑑𝑖𝑢𝑠 × 0.6 Once the wavelength is known, it is possible to calculate the velocity of the sound using the frequency of the generator by the equation: v=fλ 75  Procedures  Adjust the frequency of (350,….,… Hz). Volume may be decreased or increased as needed, by adjusting the Amplitude.  Increase the air column length (by lowering the water level) until you hear a sharp increase in volume. When you hear the increased volume, stop the water from continuing to lower. This is the first resonance position and corresponds to 1⁄4 λ.  Next, lower the water level until resonance occurs a second time. This position is about three times further down the tube and corresponds to 3/4 λ.  Repeat these steps for different frequencies. Result Resonance tube L =L1+ECF Diameter of Resonance Tube (m) = Radius of Resonance Tube (m) = ECF (m) = Frequency Column l1-l2 76 Length L L = l + ECF (m) λ = 4* L v=fλ 77 78